Interpreting Middle School Students' Proportional Reasoning Strategies: Observations From Preservice Teachers

This paper reports the findings of an investigation of 11 preservice secondary school teachers' interpretations of the development of proportional reasoning strategies used by middle school students. The preservice teachers examined samples of solution strategies generated by middle school students in proportional reasoning situations and prepared written responses of their views concerning the developmental levels indicated in the students' work. Each preservice teacher also participated in an hour‐long interview, in which the researchers asked for elaboration and clarification of the written responses and, in some cases, challenged the preservice teachers to consider alternative interpretations for the middle school students' work. The interviews were audiotaped for later analysis by the investigators, and key aspects of both the written and audiotaped responses were entered into a spreadsheet and later tabulated into categories indicating trends in the preservice teachers' interpretations. Some implications for the preparation of preservice middle school science and mathematics teachers are included.     

Assessing the Impact of Bridging Analogies in Mechanics

The effects of bridging analogies teaching strategy and gender on Turkish high school students' misconceptions in mechanics were investigated. After a pilot study with 67 students in a nearby high school, the researchers administered the revised Mechanics Misconception Test to 119 high school students as a pretest. Students in the experimental group were instructed by using bridging analogies teaching strategy. At the end of a 3‐week treatment period, the same test was administered to all students as a posttest. The data were analyzed by using analysis of covariance (ANCOVA). The statistical results showed that bridging analogies teaching strategy was an effective means of reducing the number of misconceptions students held about normal forces, frictional forces, tension, gravity, inertia, and Newton's third law.     

The Contribution of Conceptual Change Texts Accompanied by Concept Mapping to Students' Understanding of the Human Circulatory System

The present study was conducted to investigate the contribution of conceptual change texts accompanied by concept mapping instruction to 10^th— grade students' understanding of the human circulatory system. To determine misconceptions concerning the human circulatory system, 10 eleventh-grade students were interviewed. In the light of the findings obtained from student interviews and related literature, the Human Circulatory System Concepts Test was developed. The data were obtained from 26 students in the experimental group taught with the conceptual change texts accompanied by concept mapping, and 23 students in the control group taught with the traditional instruction. Besides treatment, previous learning in biology and science process skills were other independent variables involved in this study. Multiple Regression Correlation analysis revealed that science process skill, the treatment, and previous learning in biology each made a statistically significant contribution to the variation in students' understanding of the human circulatory system. It was found that the conceptual change texts accompanied by concept mapping instruction produced a positive effect on students' understanding of concepts. The mean scores of experimental and control groups showed that students in the experimental group performed better with respect to the human circulatory system. Item analysis was carried out to determine and compare the proportion of correct responses and misconceptions of students in both groups. The average percent of correct responses of the experimental group was 59.8%, and that of the control group was 51.6% after treatment.     

Polynomial calculus: rethinking the role of calculus in high schools

Access to advanced study in mathematics, in general, and to calculus, in particular, depends in part on the conceptual architecture of these knowledge domains. In this paper, we outline an alternative conceptual architecture for elementary calculus. Our general strategy is to separate basic concepts from the particular advanced techniques used in their definition and exposition. We develop the beginning concepts of differential and integral calculus using only concepts and skills found in secondary algebra and geometry. It is our underlining objective to strengthen students' knowledge of these topics in an effort to prepare them for advanced mathematics study. The purpose of this reconstruction is not to alter the teaching of limit-based calculus but rather to affect students' learning and understanding of mathematics in general by introducing key concepts during secondary mathematics courses. This approach holds the promise of strengthening more students' understanding of limit-based calculus and enhancing their potential for success in post-secondary mathematics.     

Students' Concepts‐ and Theorems‐in‐Action on a Novel Task About Similarity

Similarity is a fundamental concept in the middle grades. In this study, we applied Vergnaud's theory of conceptual fields to answer the following questions: What concepts‐in‐action and theorems‐in‐action about similarity surfaced when students worked in a novel task that required them to enlarge a puzzle piece? How did students use geometric and multiplicative reasoning at the same time in order to construct similar figures? We found that students used concepts of scaling and proportional reasoning, as well as the concept of circle and theorems about similar triangles, in their work on the problem. Students relied not only on visual perception, but also on numeric reasoning. Moreover, students' use of multiplicative and proportional concepts supported their geometric constructions. Knowledge of the concepts and ideas that students have available when working on a task about similarity can inform instruction by helping to ground formal introduction of new concepts in students' informal prior experiences and knowledge.     

Two Teaching Methods and Students' Understanding of Sound

The purpose of this study was to examine fifth grade students' ideas related to sound and to compare the Learning Cycle teaching approach with a textbook/demonstration method of instruction to determine whether one method is more effective in facilitating conceptual change. Thirty-four fifth grade students were randomly selected and assigned to the two treatment groups. To assess the students' understanding of specific sound concepts, an interview protocol was administered to both groups before and immediately after instruction. Students were given a numerical rating corresponding to their levels of understanding. The numerical values for both groups at the pre- and post-interview assessments were analyzed by analysis of variance (ANOVA). Students who were taught using the Learning Cycle had a significantly better understanding.     

Fuzzy Logic,Neural Networks,Genetic Algorithms: Views of Three Artificial Intelligence Concepts Used in Modeling Scientific Systems

Students' conceptions of three major artificial intelligence concepts used in the modeling of systems in science, fuzzy logic, neural networks, and genetic algorithms were investigated before and after a higher education science course. Students initially explored their prior ideas related to the three concepts through active tasks. Then, laboratories, project work, use of computer modeling of scientific systems, and cooperative group work were used to help students construct key characteristics of each concept. Finally, they applied each concept in contexts different from that in which it had been previously studied. In postcourse interviews using a set of scenarios for each of the major course concepts, 49% of students' applications included key characteristics of the concepts studied versus an application of 5% in precourse interviews. Students' post interview applications were inconsistent even though they were more frequent, indicating a state of transition in their conceptual change. Applications were most consistent when used with scenarios deemed very familiar to the students, indicating the effects of context in conceptual change.     

Understanding and representing the arithmetic averaging algorithm: an analysis and comparison of US and Chinese students' responses

Analysing the responses of 311 sixth-grade Chinese students and 232 sixth-grade US students to two problems involving arithmetic average, this study explored students' understanding and representation of the averaging algorithm from a cross-national perspective. Results of the study show that Chinese students were more successful than US students in obtaining correct numerical answers to each of the problems, but US and Chinese students had similar cognitive difficulties in solving the second task. The difficulties were not due to their lack of procedural knowledge of the averaging algorithm, rather due to their lack of conceptual understanding of the algorithm. There were significant differences between the US and Chinese students in their solution representations of the two average problems. Chinese students were more likely to use algebraic representations than US students; while US students were more likely to use pictorial or verbal representations. US and Chinese students' use of representations are related to their mathematical problem-solving performance. Students who used more advanced representations were better problem solvers. The findings of the study suggest that Chinese students' superior performance on the averaging problems is partly due to their use of advanced representations (e.g. algebraic).     

Students' Understanding of the Particulate Nature of Matter

The particulate nature of matter is identified in science education standards as one of the fundamental concepts that students should understand at the middle school level. However, science education research in indicates that secondary school students have difficulties understanding the structure of matter. The purpose of the study is to describe how engaging in an extended project‐based unit developed urban middle school students' understanding of the particulate nature of matter. Multiple sources of data were collected, including pre‐ and posttests, interviews, students' drawings, and video recordings of classroom activities. One teacher and her five classes were chosen for an indepth study. Analyses of data show that after experiencing a series of learning activities the majority of students acquired substantial content knowledge. Additionally, the finding indicates that students' understanding of the particulate nature of matter improved over time and that they retained and even reinforced their understanding after applying the concept. Discussions of the design features of curriculum and the teacher's use of multiple representations might provide insights into the effectiveness of learning activities in the unit.     

Developing Student Understanding: Contextualizing Calculus Concepts

This study looked at the practice of one high school teacher who provided students with concrete examples from their physics class to give them a contextually rich environment in which to explore the abstractions of calculus. Students discovered connections between the physics concepts of position, velocity, and acceleration and the calculus concepts of function, derivative, and antiderivative. The qualitative study sought to describe several critical aspects of understanding: students' ability to explain concepts and procedures, to apply concepts in a physics context, and to explore their own learning. It included 32 seniors at a large, urban, comprehensive, religious school in a midwestern stale. Samples of student work and reflections were collected by the teacher, as well as by students in individual portfolios. The teacher kept a reflective journal. This study suggests that making connections between calculus and physics can yield deep understandings of semantic as well as procedural knowledge.     

High School Biology Students' Knowledge and Certainty about Diffusion and Osmosis Concepts

The purpose of this study was to investigate students' understanding about scientifically acceptable content knowledge by exploring the relationship between knowledge of diffusion and osmosis and the student's certainty in their content knowledge. Data was collected from a high school biology class with the Diffusion and Osmosis Diagnostic Test (DODT) and Certainty of Response (CRI) scale. All data was collected after completion of a unit of study on diffusion and osmosis. The results of the DODT were dichotomized into correct and incorrect answers, and CRI values were dichotomized into certain and uncertain. Values were used to construct a series of 2 × 2 contingency tables for each item on the DODT and corresponding CRI. High certainty in incorrect answers on the DODT indicated tenacious misconceptions about diffusion and osmosis concepts. Low certainty in incorrect or correct answers on the DODT indicated possible guessing; and, therefore no understanding, or confusion about their understanding. Chi‐square analyses revealed that significantly more students had misconceptions than desired knowledge on content covering the Influence of Life Forces on Diffusion and Osmosis, Membranes, the Particulate and Random Nature of Matter, and the Processes of Diffusion and Osmosis. Most students were either guessing or had misconceptions about every item related to the concepts osmosis and tonicity. Osmosis and diffusion are important to understanding fundamental biology concepts, but the concept of tonicity not be introduced to high school biology students until effective instructional approaches can be identified by researchers.     

The Effects of Non‐CAS Graphing Calculators on Student Achievement and Attitude Levels in Mathematics: A Meta‐Analysis

Forty‐two studies comparing students with access to graphing calculators during instruction to students who did not have access to graphing calculators during instruction are the subject of this meta‐analysis. The results on the achievement and attitude levels of students are presented. The studies evaluated cover middle and high school mathematics courses, as well as college courses through first semester calculus. When calculators were part of instruction but not testing, students' benefited from using calculators while developing the skills necessary to understand mathematics concepts. When calculators were included in testing and instruction, the procedural, conceptual, and overall achievement skills of students improved.     

Turkish high school students’ definitions for parallelograms: appropriate or inappropriate?

The aim of this study was to investigate the appropriateness of high school students' definitions. The participants in this study were 269 high school students from a public school in Ordu city, which is on the Black Sea coast of Turkey. The participants were asked to write their definitions with no time constraints. In the analysis of the definitions, students' ability to distinguish necessary and sufficient conditions and their ability to use appropriate mathematical terminology were taken into account. The task used in this study enabled us to mirror students' difficulties and inadequacies about their definitions of a parallelogram. The findings indicated that most of the students defined parallelogram inappropriately because they had used incomplete or incorrect statements. On the other hand, for the appropriate definitions, it was found that the number of uneconomical definitions was almost the same as the number of economical ones. At the end of the study, it was suggested that defining activities should be integrated into curriculums explicitly and should be given importance in our mathematic lessons.     

On problematic aspects in learning trigonometry

In this paper, research on some problematic aspects high school students have in learning trigonometry is presented. It is based on making sense of mathematics through perception, operation and reason in the case of trigonometry. We analyzed students' understanding of trigonometric concepts in the frame of triangle and circle trigonometry contexts, as well as the transition between these two contexts. In the conclusion, we present some new problematic aspects we noticed.

The research was carried out with two groups of high school students, one of them at the beginning of their trigonometry learning (17 years old) and the other at the end of their high school education (19 years old). The students were given a questionnaire similar to that of Chin and Tall, and we analyzed the students' response. In our research, we noticed that students have difficulties with properties of periodicity and the fact that trigonometric functions are not one-to-one. In addition, there is poor understanding of radian measure and a lack of its connection to the unit circle.     

Prime decomposition: Understanding uniqueness

This study investigates procedural and conceptual aspects in preservice elementary school teacher's understanding of the Fundamental Theorem of Arithmetic. The data were collected by the means of a written questionnaire and individual interviews. The results suggest that the idea of the uniqueness of prime decomposition is very difficult to grasp. Participants' responses indicated, either implicity or explicity, that a possibility of alternative prime decompositions was often not overruled, and this influenced students' ability to make inferences regarding factors and divisors of natural numbers. Some pedagogical implications are discussed.     

Examining the Conceptual Organization of Students in an Integrated Algebra and Physical Science Class

The cross-disciplinary context of density and slope was used to compare the conceptual organization of students in an integrated algebra and physical science class (SAM9) with that of students in a discipline-specific physical science class (PSO). Analyses of students' concept maps indicated that the SAM9 students used a greater number of procedural linkages to connect mathematics and science concepts on the SAM9 students' maps than did the PSO students. The maps produced by SAM9 students also tended to show a more compartmentalized approach to thinking about the content of the two disciplines, a finding contrary to the researcher's original assertion. Traditional teaching territories and conceptual complexity were examined as possible explanations for the discrepancy between the predicted and actual outcomes.     

The Influence of Science Summer Camp on African‐American High School Students' Career Choices

This study explored if a weeklong science camp changed Louisiana African‐American high school students' perception of science. A semi‐structured survey was used before and after the camp to determine the changes in science attitudes and career choices. Among the perceived benefits were parental involvement, increased science academic ability, and deepened scientific knowledge. These perceived benefits influenced the identities that students constructed for themselves in relation to science in their lives. Students who reported doing well in school science courses believed that science was more relevant to their lives. Female students who cited doing well in science reported less parental involvement in their schoolwork than males. This study draws attention to gender differences in science and to designing informal science learning experiences for African‐American high school students that can change attitudes toward career choices in science‐related fields.     

Diagnosing conflict factors in calculus through students' writings: One teacher's reflections

One means of effecting innovation in calculus teaching and learning involves curricular change, but implementing a reform calculus curriculum does not mean necessarily that students go beyond rote learning and symbolic manipulation. In this paper, we document a process of self-reflection, during which the teacher-as-researcher engaged in analysis of students' written responses to prompts that were intended to inform the teacher about students' understanding of fundamental concepts.     

Applying Cognition-Based Assessment to Elementary School Students' Development of Understanding of Area and Volume Measurement

As part of a discussion of cognition-based assessment (CBA) for elementary school mathematics, I describe assessment tasks for area and volume measurement and a research-based conceptual framework for interpreting students' reasoning on these tasks. At the core of this conceptual framework is the notion of levels of sophistication. I provide details on an integrated set of levels for area and volume measurement that (a) starts with the informal, preinstructional reasoning typically possessed by students, (b) ends with the formal mathematical concepts targeted by instruction, and (c) indicates cognitive plateaus reached by students in moving from (a) to (b).     

From formal standards to everyday practice of mathematics learning

This paper uses the example of six Japanese teachers and their mathematics lessons to illustrate how clear, high standards for mathematics instruction are combined with teachers' holistic concern for students. We draw upon data from the Third International Math and Science Study Case Study Project in Japan that was designed to elucidate the context behind the high achievement of Japanese students. Using everyday examples of classroom practice, we illustrate both flexibility in teachers' approach to teaching and adherence to Monbusho's (Ministry of Education, Science, Sports, and Culture)Course of Study. Our purpose is to emphasize how flexibility and attention to individual needs by Japanese teachers combine with quality mathematics instruction based on the detailed Japanese curricula. Six teachers' characteristics and lessons (two teachers at each educational level—elementary, junior high, and high school) are described in order to show the variety of teachers who exist in Japan. These teachers use their understanding of theCourse of Study and are supported by their school environment to enhance their students' conceptual understanding of the fundamentals of mathematics. Characteristics of their teaching include: 1) involving the whole class in learning. 2) using extremely focused curriculum guidelines that expect mastery of concepts at each grade level, 3) thoroughly covering mathematics units in an organized and in-depth manner, 4) leading classes as facilitators or guides more often than as lecturers, and 5) focusing on problem solving with the primary goal of developing students' ability to reason, especially to reason inductively. The examples in this paper show how these methods develop in individal classrooms.